All We Need is Loève: Transfer Function Estimation Between Non-Stationary Processes

The study of geomagnetism has been documented as far back as 1722 (Graham) with increased interest at the end of the 19th century (Lamb, Schuster, Chapman, and Price). The Magnetotelluric Method was first introduced in the 1950's (Cagniard and Tikhonov), and, at its core, is simply a regression problem. The result of this method is a transfer function estimate that describes the earth's response to magnetic field variations. This estimate can then be used to infer the earth's subsurface structure; useful for applications such as natural resource exploration, or to model the effects of magnetic field variations on a power transmission system, pipeline, or other long conductor.

The statistical problem of estimating a transfer function between geomagnetic and induced current measurements has evolved since the 1950's due to a variety of problems: non-stationarity, outliers, and violation of Gaussian assumptions. To address some of these issues, robust regression methods (Chave and Thomson, 2004) and the remote reference method (Gambel, 1979) have been proposed and used. The current transfer function models provide reasonable results, but the estimates only model the relationship (H) between the same frequencies in input (X) and response (Y), i.e. have the form:

Using the multitaper method of spectral analysis (Thomson, 1982), taking long (greater than 3 months) blocks of geomagnetic data, and concentrating on frequencies below 1000 microhertz to avoid ultraviolet effects, one finds that:
1) the cross-spectra are dominated by many offset frequencies including plus and minus 1 and 2 cycles per day and
2) the coherence at these offset frequencies is often stronger than at zero offset.

Due to the non-stationary nature of the series, two variables are required to represent the covariance between the geomagnetic and geoelectric processes (Loève, 1946). This implies that multiple frequencies in the input series should also be used:

We discuss this model, challenges, and current results.